Based on the classic paper by Holling in 1959, predator-prey interaction can be modeled by the following functions (i.e. so called Type I or Type II functional response). Both functional responses contains a parameter that describes the proportion of prey that is available for the predator.
\[f(R) = {\alpha}TR \], where \({\alpha}\) is the encounter probability, \(T\) is the searching time, and \(R\) is the prey density. Encounter probability (\({\alpha}\)) is the proportion of prey that is available for the predator.
\[f(R) = \frac{{\alpha}TR}{1+{\alpha}hR}\], where \({\alpha}\) is again the encounter probability, \(T\) is the searching time, and \(h\) is the handling time.
In my lab experiment, I directly modified the \({\alpha}\) parameter because I used the screen mesh to modify the proportion of IG prey that is available to the IG predator. This lab experiment would thus be valid to verify my model predictions.
First visualize the population dynamics of the two protozoa species.
Blepharisma
Colpidium
For both Blespharisma and Colpidium, I calculated the mean density for all six treatments (0%, 20%, 40%, 60%, 80% and 100% encounter probability) at the steady state. Because I cannot be 100% sure what are the most representative time points for the steady state, I calculate the mean density of every five consecutive time points. I then moved this five-point time frame gradually forward from hour 34 to hour 535. Finally , I chose the time frame of hour 320 to 486 since the other time frames are qualitatively the same.
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## Kruskal-Wallis rank sum test
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## data: avg by as.factor(trmt.all)
## Kruskal-Wallis chi-squared = 19.688, df = 5, p-value = 0.00143
## Kruskal-Wallis rank sum test
##
## data: x and group
## Kruskal-Wallis chi-squared = 19.6877, df = 5, p-value = 0
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##
## Comparison of x by group
## (Benjamini-Hochberg)
## Col Mean-|
## Row Mean | C00 C02 C04 C06 C08
## ---------+-------------------------------------------------------
## C02 | 2.247567
## | 0.0461
## |
## C04 | 2.643139 0.395571
## | 0.0308 0.4328
## |
## C06 | 4.387252 2.139684 1.744112
## | 0.0001* 0.0347 0.0676
## |
## C08 | 2.013820 -0.233747 -0.629318 -2.373431
## | 0.0413 0.4703 0.3969 0.0441
## |
## C10 | 2.193626 -0.053941 -0.449513 -2.193626 0.179805
## | 0.0353 0.4785 0.4453 0.0424 0.4593
For the Colpidium, We see that 0% encounter probability treatment, which is the treatment where two protozoa species engage in only competition, has significantly higher density than the other trThis suggests that Colpidium density would be lower as long as intra-guild predation occurs.
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## Kruskal-Wallis rank sum test
##
## data: avg by as.factor(trmt.all)
## Kruskal-Wallis chi-squared = 9.1165, df = 5, p-value = 0.1045
## Kruskal-Wallis rank sum test
##
## data: x and group
## Kruskal-Wallis chi-squared = 9.1145, df = 5, p-value = 0.1
##
##
## Comparison of x by group
## (Benjamini-Hochberg)
## Col Mean-|
## Row Mean | B00 B02 B04 B06 B08
## ---------+-------------------------------------------------------
## B02 | 0.559584
## | 0.3599
## |
## B04 | 2.021724 1.462140
## | 0.1080 0.1796
## |
## B06 | 2.166133 1.606549 0.144408
## | 0.1136 0.1622 0.4426
## |
## B08 | 2.328593 1.769009 0.306868 0.162460
## | 0.1491 0.1442 0.4379 0.4666
## |
## B10 | 1.155271 0.595686 -0.866453 -1.010862 -1.173322
## | 0.2325 0.3759 0.2897 0.2601 0.2579
Density of IG predator at the steady state are not significanly different from each other.
Now I check the bacteria consumption in different treatments. To calculate bacteria consumption, I first calculate the bacteria density change in each treatment, including the control treatment where two protozoa species are not present. I then compare the bacteria density change in each treatment (i.e. 0%, 20%, 40%, 60%, 80%, 100% encounter probability) to the control treatment to calculate the bacteria consumption in each treatment.
According to the mathematical model, I hypothesize that bacteria consumption would first increase and than decrease with intra-guild predation rate (see below). Again, I used the encounter probability between the two protozoa species to represent the intra-guild predation rate.